IMHO, the same situation is for Godel numbers. We can use math to generate numbers, which are equivalent to formulas, BUT why these formulas must be accepted? Initial set of axioms was carefully chosen by human, so it not an open set, where anybody can add anything.
The formulae are accepted because Gödel gives us a mapping, he proves that we can convert any formula into a number without losing information.
However, some formulas are incorrect, so we filter out them and then remap remaining functions again: 1 is first correct formula, 2 is second correct formula, and so on. Now, we can produce correct formulas with just «next» operator.
However, some formulas can contradict our system of rules, so we filter out them, and remap remaining functions again: 1 is the first correct formula which doesn't contradict the system of rules, and so on. Let's call them "Lisivka's numbers".
So, Godel's numbers can contradict axiomatic system, while Lisivka's numbers cannot.
Do you see the problem?
Try going through the essay, and point out where the “incorrect” numbers were formed. You may be surprised to find that all statements were “correct” in the definition you are thinking of. The mathematical term is “primitive recursive” and “well-formed”
> We can go further. We can even construct PM-Lisp formulas in PM-Lisp!
No, we cannot.
To see how it feels:
No, drran, you have missed the point. Read it again, maybe you'll get it.
This number is equivalent to the proof that I'm right. Godel was genius!
Mathematica does not correctly represent the proposition
because it leaves out the order of the proposition.